Let's address the problem step by step.

Part (a): Prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}$.

We are given that $z$ and $w$ are complex numbers such that $|z| = |w| = 1$, meaning both numbers lie on the unit circle in the complex plane. The modulus of a complex number $z$ is given by $|z| = \sqrt{z\overline{z}}$, and since $|z| = 1$, this gives: $|z|^2 = z\overline{z} = 1.$ Thus, we can solve for $\overline{z}$: $\overline{z} = \frac{1}{z}.$ Similarly, for $w$, we have: $|w|^2 = w\overline{w} = 1,$ so $\overline{w} = \frac{1}{w}.$ This proves part (a).

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Part (b): Prove that $\frac{z + w}{zw + 1}$ is a real number.

To show that $\frac{z + w}{zw + 1}$ is real, we need to demonstrate that the imaginary part of this expression is zero.

We will use the fact that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}$, as well as the property of conjugates that for a complex number $z$, if $z$ is real, then $\overline{z} = z$.

The conjugate of the expression $\frac{z + w}{zw + 1}$ is given by: $\overline{\left( \frac{z + w}{zw + 1} \right)} = \frac{\overline{z + w}}{\overline{zw + 1}}.$ Using the properties of conjugation, we can distribute the conjugate over the sum and product: $\overline{z + w} = \overline{z} + \overline{w} = \frac{1}{z} + \frac{1}{w},$ and $\overline{zw + 1} = \overline{zw} + \overline{1} = \overline{z} \cdot \overline{w} + 1 = \frac{1}{z} \cdot \frac{1}{w} + 1 = \frac{1}{zw} + 1.$ Thus, the conjugate of the expression becomes: $\frac{\frac{1}{z} + \frac{1}{w}}{\frac{1}{zw} + 1}.$ Now, simplify $\frac{1}{z} + \frac{1}{w}$ and $\frac{1}{zw} + 1$: $\frac{1}{z} + \frac{1}{w} = \frac{w + z}{zw},$ and $\frac{1}{zw} + 1 = \frac{1 + zw}{zw}.$ Substitute these into the conjugate expression: $\overline{\left( \frac{z + w}{zw + 1} \right)} = \frac{\frac{z + w}{zw}}{\frac{1 + zw}{zw}} = \frac{z + w}{1 + zw}.$ This shows that: $\overline{\left( \frac{z + w}{zw + 1} \right)} = \frac{z + w}{zw + 1}.$ Since the expression equals its conjugate, it must be a real number.

Thus, we have shown that $\frac{z + w}{zw + 1}$ is a real number, which completes the proof of part (b).

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Would you like more details or further explanation?

Here are 5 questions to further explore related topics:

1. What are the properties of complex conjugates on the unit circle?
2. How do conjugation and modulus relate to real and imaginary parts of complex numbers?
3. What conditions must hold for a complex fraction to be real?
4. How can we interpret the geometric meaning of $|z| = 1$ in the complex plane?
5. Can you generalize the result to other complex functions of two variables?

Tip: When working with complex numbers, conjugates and moduli are essential tools to manipulate and simplify expressions, particularly when dealing with unit circle properties.